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Mirrors > Home > ILE Home > Th. List > m1expeven | GIF version |
Description: Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
Ref | Expression |
---|---|
m1expeven | ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 8753 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | 1 | 2timesd 8656 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · 𝑁) = (𝑁 + 𝑁)) |
3 | 2 | oveq2d 5668 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = (-1↑(𝑁 + 𝑁))) |
4 | neg1cn 8525 | . . . 4 ⊢ -1 ∈ ℂ | |
5 | neg1ap0 8529 | . . . 4 ⊢ -1 # 0 | |
6 | expaddzap 9995 | . . . 4 ⊢ (((-1 ∈ ℂ ∧ -1 # 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) | |
7 | 4, 5, 6 | mpanl12 427 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
8 | 7 | anidms 389 | . 2 ⊢ (𝑁 ∈ ℤ → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁))) |
9 | m1expcl2 9973 | . . 3 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | |
10 | neg1rr 8526 | . . . . . 6 ⊢ -1 ∈ ℝ | |
11 | reexpclzap 9971 | . . . . . 6 ⊢ ((-1 ∈ ℝ ∧ -1 # 0 ∧ 𝑁 ∈ ℤ) → (-1↑𝑁) ∈ ℝ) | |
12 | 10, 5, 11 | mp3an12 1263 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℝ) |
13 | elprg 3466 | . . . . 5 ⊢ ((-1↑𝑁) ∈ ℝ → ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1))) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) ∈ {-1, 1} ↔ ((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1))) |
15 | oveq12 5661 | . . . . . . 7 ⊢ (((-1↑𝑁) = -1 ∧ (-1↑𝑁) = -1) → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) | |
16 | 15 | anidms 389 | . . . . . 6 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = (-1 · -1)) |
17 | neg1mulneg1e1 8626 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
18 | 16, 17 | syl6eq 2136 | . . . . 5 ⊢ ((-1↑𝑁) = -1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
19 | oveq12 5661 | . . . . . . 7 ⊢ (((-1↑𝑁) = 1 ∧ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) | |
20 | 19 | anidms 389 | . . . . . 6 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = (1 · 1)) |
21 | 1t1e1 8566 | . . . . . 6 ⊢ (1 · 1) = 1 | |
22 | 20, 21 | syl6eq 2136 | . . . . 5 ⊢ ((-1↑𝑁) = 1 → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
23 | 18, 22 | jaoi 671 | . . . 4 ⊢ (((-1↑𝑁) = -1 ∨ (-1↑𝑁) = 1) → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
24 | 14, 23 | syl6bi 161 | . . 3 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) ∈ {-1, 1} → ((-1↑𝑁) · (-1↑𝑁)) = 1)) |
25 | 9, 24 | mpd 13 | . 2 ⊢ (𝑁 ∈ ℤ → ((-1↑𝑁) · (-1↑𝑁)) = 1) |
26 | 3, 8, 25 | 3eqtrd 2124 | 1 ⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 664 = wceq 1289 ∈ wcel 1438 {cpr 3447 class class class wbr 3845 (class class class)co 5652 ℂcc 7346 ℝcr 7347 0cc0 7348 1c1 7349 + caddc 7351 · cmul 7353 -cneg 7652 # cap 8056 2c2 8471 ℤcz 8748 ↑cexp 9950 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3394 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-ilim 4196 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-frec 6156 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-inn 8421 df-2 8479 df-n0 8672 df-z 8749 df-uz 9018 df-iseq 9849 df-seq3 9850 df-exp 9951 |
This theorem is referenced by: m1expe 11173 m1expo 11174 m1exp1 11175 |
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